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Adjusted Significance Level of Blinded Sample Size Re-Estimation for Paired Binary Data

Source: R/PairedAlphaAdjBSSR.R
PairedAlphaAdjBSSR.Rd

Lowers the nominal significance level of a matched-pairs design with blinded sample size re-estimation until the largest type I error rate over the probability of a discordant pair on the boundary of the null hypothesis does not exceed the target level alpha, and returns the adjusted level together with that of the fixed design.

Usage

PairedAlphaAdjBSSR(
  Delta.A,
  N,
  omega = NULL,
  alpha,
  tar.power,
  Test,
  restricted = FALSE,
  alternative = c("greater", "less", "two.sided", "equivalence"),
  n.grid = 100,
  bb.gamma = 0,
  ss.method = c("exact", "standard", "conditional.power"),
  ss.Test = Test,
  ss.alpha = alpha,
  search = c("crossing", "smallest", "stable"),
  search.limit = c(2, 50),
  N.min = NULL,
  N.max = NULL,
  n.interim = NULL,
  psi = seq(0, 1, by = 0.01),
  maximize = c("certified", "refined", "grid"),
  adjust = c("test", "both"),
  tol = 1e-08,
  step = 1e-05,
  margin = 0,
  ref.pvalue = FALSE
)

Arguments

Delta.A

Assumed difference p10 - p01 used in the re-estimation

N

Planned number of pairs, which is the number of pairs of the fixed design

omega

Fraction of N at the interim analysis, rounded up. Supply either omega or n.interim

alpha

Target level of significance, which the type I error rate must not exceed

tar.power

Target power of the re-estimation

Test

Type of statistical test. Options: 'McNemar', 'McNemar-cc', 'McNemar-exact', 'McNemar-midP', 'McNemar-uncond' or 'Paired-Score'

restricted

Logical. If TRUE, the final number of pairs is at least N. Default is FALSE

alternative

Direction of the alternative hypothesis. Options: 'greater' (default) for p10 - p01 > -margin, 'less' for p10 - p01 < margin, 'two.sided' for p10 != p01, or 'equivalence' for -lower < p10 - p01 < upper

n.grid

Number of grid points used to search over the probability of a discordant pair in the test 'McNemar-uncond'. Default is 100

bb.gamma

Confidence level parameter of the Berger-Boos procedure for the test 'McNemar-uncond'. The default of 0 disables the procedure

ss.method

Re-estimation rule. 'exact' (default) and 'standard' re-estimate the number of pairs of PairedSampleSize with the method of that name. 'conditional.power' takes the smallest number of pairs at which the conditional power given the interim number of discordant pairs attains tar.power, the rule of Schepers et al. (2026); it requires N.max. See Details

ss.Test

Test used in the re-estimation. Default is Test

ss.alpha

Level of significance used for the re-estimation when adjust = 'test'. Default is alpha. It is ignored when adjust = 'both'

Search of the exact re-estimation, see PairedSampleSize; it does not go beyond N.max

search.limit

Limit of the search 'stable', see PairedSampleSize

N.min

Smallest final number of pairs, or NULL

N.max

Largest final number of pairs, or NULL

n.interim

Number of pairs at the interim analysis. Supply either omega or n.interim

psi

Values of the probability p10 + p01 of a discordant pair at which the rate is evaluated. Values that do not lie on the boundary of the null hypothesis, below |margin|, are dropped. Default is seq(0, 1, by = 0.01)

maximize

How the largest type I error rate at a level is located, as in PairedTypeIErrorBSSR. With 'certified' (default) the adjusted level controls the type I error rate over the whole interval spanned by the values of psi on the boundary. With 'refined' and 'grid' the control is assessed on the grid

adjust

Which part of the design uses the adjusted level. 'test' (default) applies it to the final analysis only, so the re-estimation keeps using ss.alpha. 'both' applies it to the re-estimation as well

tol

Tolerance of the bisection, relative to alpha, a single value in (0, 1). Default is 1e-8

step

Decrement of the level in the search used when adjust = 'both'. Default is 1e-5

margin

Margin on the scale of the difference p10 - p01. The default of 0 gives a test of superiority, and a value other than 0 tests the null hypothesis p10 - p01 <= -margin against p10 - p01 > -margin when alternative is 'greater', and p10 - p01 >= margin against p10 - p01 < margin when it is 'less'. A negative value tests for superiority by more than its absolute value. A margin other than 0 requires Test = 'Paired-Score' and a one-sided alternative. For alternative = 'equivalence' the margin is the pair c(lower, upper), both in (0, 1), or a single value used for both, and the null hypothesis p10 - p01 <= -lower or p10 - p01 >= upper is tested

ref.pvalue

Logical. If TRUE, the maximization over the probability of a discordant pair in the test 'McNemar-uncond' is refined between the grid points, see Details. Default is FALSE

Value

An object of class bbssr_alphaadj, a data frame with one row for the BSSR design and one for the fixed design containing:

Design

'BSSR' or 'TRAD'

alpha

Target level

max.TIE

Largest type I error rate at the nominal level alpha

alpha.adj

Adjusted nominal level. It is alpha when the nominal level already controls the type I error rate, and 0, with a warning, when no level examined by the bisection does. The search of adjust = 'both' for the BSSR design stops with an error instead when it reaches 0

max.TIE.adj

Largest type I error rate at the adjusted level

psi.adj

Probability of a discordant pair at which max.TIE.adj occurs, NA when alpha.adj is 0

max.TIE.bound

Upper bound of the type I error rate at the nominal level with maximize = 'certified', otherwise NA

max.TIE.adj.bound

Upper bound of the type I error rate at the adjusted level with maximize = 'certified', otherwise NA

For alternative = 'equivalence' the column boundary.adj gives the boundary, 'lower' or 'upper', on which max.TIE.adj occurs.

Details

The searches are those of BinaryAlphaAdjBSSR. With adjust = 'test' the final numbers of pairs do not depend on the nominal level of the final analysis, the largest type I error rate is a non-decreasing function of the level, and the level is found by bisection to within tol * alpha; the reported level is the largest value with at most six significant digits that gives the same rejection regions. With adjust = 'both' the level is lowered from alpha in steps of step until the type I error rate is controlled. The fixed design always uses the bisection. With maximize = 'certified' a level is accepted only when the upper bound of the type I error rate over the interval, computed as in PairedTypeIErrorBSSR, does not exceed alpha.

The exact conditional McNemar test keeps its level whatever blinded rule is used (Schepers et al., 2026), so it needs no adjustment.

References

Kieser M, Friede T (2000). Re-calculating the sample size in internal pilot study designs with control of the type I error rate. Statistics in Medicine, 19(7), 901-911.

Schepers M, Brannath W, Hoffmann E, Stingl J, Schmidtmann I (2026). Blinded sample size review for McNemar's test based on primary and surrogate endpoints. arXiv:2608.17784.

Examples

# \donttest{
PairedAlphaAdjBSSR(Delta.A = 0.2, N = 40, n.interim = 20, alpha = 0.05,
                   tar.power = 0.8, Test = 'McNemar', alternative = 'two.sided',
                   ss.method = 'standard', psi = seq(0.05, 0.95, by = 0.05))
#> Adjusted significance level for blinded sample size re-estimation
#> 
#>   Test            : McNemar
#>   Alternative     : two.sided
#>   Adjusted part   : final analysis only
#>   Maximum         : certified over psi in [0.05, 0.95]
#>   Target level    : 0.05
#> 
#>        Design   max.TIE alpha.adj max.TIE.adj
#>          BSSR 0.0505829 0.0496472   0.0495041
#>  Fixed sample 0.0538321 0.0481934   0.0485610
# }